Quadratic factored form
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- [Instructor] We're asked to graph the equation y is equal to one-half times x minus six times x plus two. So like always, pause this video and take out some graph paper or even try to do it on a regular piece of paper and see if you can graph this equation. All right now, let's work through this together. There's many different ways that you could attempt to graph it. May be the most basic is try out a bunch of x values and a bunch of y values and try to connect the curve that connects all of those dots. But let's try to see if we can get the essence of this graph without doing that much work. The key realization here without even having to do the math is if I multiply this out, if I multiplied x minus six times x plus two, I'm going to get a quadratic. I'm going to get x-squared, plus something, plus something else. And so this whole thing is going to be a parabola. We are graphing a quadratic equation. Now a parabola you might remember can intersect the x-axis multiple times. So let's see if we can find out where this intersects the x-axis. And the form that it's in, it's in factored form already, it makes it pretty straightforward for us to recognize when does y equal zero? Which are going to be the times that we're intersecting the x-axis. And then from that, we'll actually be able to find the coordinates of the vertex and we're going to be able to get the general shape of this curve which is going to be a parabola. So let's think about it. When does y equal zero? Well to solve that we just have to figure out when, if we want to know when y equals zero, then we have to solve for when does this expression equal zero? So let's just solve the equation. One half times x minus six, times x plus two is equal to zero. Now in previous videos we've talked about this idea. If I have the product of multiple things and it needs to be equal to zero, the only way that's going to happen is if one or more of these things are going to be equal to zero. Well one half is one-half, it's not going to be equal to zero. But x minus six could be equal to zero. So if x minus six is equal to zero, then that would make this equation true. Or if x plus two is equal to zero, that would also make this equation true. So the x values that satisfy either of these would make y equal zero and those would be places where our curve is intersecting the x-axis. So what x value makes x minus six equal zero? Well you could add six to both sides, you're probably able to do that in your head, and you get x is equal to six. Or you subtract two from both sides here and you get x is equal to, these cancel out, you get x is equal to negative two. These are the two x values where y will be equal to zero. You can substitute it back into our original equation. If x is equal to six, then this right over here is going to be equal to zero, and then y is going to be equal to zero. If x is equal to negative two, then this right over here is going to be equal to zero, and y would be equal to zero. So we know that our parabola is going to intersect the x-axis at x equals negative two right over there, and x is equal to six. These are our x-intercepts. So given this, how do we figure out the vertex? Well the key idea here is to recognize that your axis of symmetry for your parabola is going to sit right between your two x-intercepts. So what is the midpoint between, or what is the average of six and negative two? Well, you could do that in your head. Six plus negative two is four divided by two is two. Let me do that. So I'm just trying to find the midpoint between the point, let's use a new color. So I'm trying to find the midpoint between the point negative two comma zero and six comma zero. Well the midpoint, those are just the average of the coordinates. The average of zero and zero is just going to be zero, it's going to sit on the x-axis. But then the midpoint of negative two and six or the average negative two plus six over two. Well let's see, that's four over two, that's just going to be two, so two comma zero. And you see that there. You could have done that without even doing the math. You say okay, if I want to go right in between the two, I want to be two away from each of them. And so just like that, I could draw an axis of symmetry for my parabola. So my vertex is going to sit on that axis of symmetry. And so how do I know what the y value is? Well I can figure out, I can substitute back in my original equation, and say well what is y equal when x is equal to two? Because remember the vertex has a coordinate x equals two. It's going to be two comma something. So let's go back, let's see what y equals. So y will equal to one-half times, we're going to see when x equals two, so two minus six, times two plus two. Let's see, this is negative four, this is positive four. Negative four times four is negative 16. So it's equal to one-half times negative 16, which is equal to negative eight. So our vertex when x is equal to two, y is equal to negative eight. And so our vertex is going to be right over here two common negative eight. And now we can draw the general shape of our actual parabola. It's going to look something like, once again this is a hand-drawn sketch, so take it with a little bit of a grain of salt, but it's going to look something like this. And it's going to be symmetric around our axis of symmetry. That's why it's called the axis of symmetry. This art program I have, there's a symmetry tool, but I'll just use this and there you go. That's a pretty good sketch of what this parabola is or what this graph is going to look like which it is, an upward-opening parabola.